posted by Kelsey .
A rectangular sheet of paper of width a and length b, where 0<a<b, is folded by taking one corner of the sheet and placing it at point P on the opposite long side of the sheet. The fold is flatened to form a crease across the sheet. Assuming that the fold is made that there is no flap extending beyond the original sheet, find the point P that produces the crease of minimum length. What is the length of that crease?
If we label the lower left corner A, and go clockwise for B,C,D
then with some actual folding experimentation, we see that P can go along side BC at a distance x from B, which can vary from b-sqrt(b^2-a^2) to a. Anywhere else the fold produces a flap that extends beyond the sheet.
When x = b-sqrt(b^2-a^2) the fold length is just the long side of the rectangle, b. That is, P is a distance sqrt(b^2-a^2) from C.
When x = a, the fold length is just the diagonal of a square, a√2
It's late. I'll let you figure out in which cases one is greater than the other.